LIQUID SPHEROID OF FINITE ELLIPTICITY. 
197 
M = [^0 + + /)] + M + K + ?r)} + const. (44), 
where the square brackets indicate that C is to be put equal to Co¬ 
la the case of forced tides due partly to small disturbing forces whose potential at 
any instant is Vg, and partly to periodic variations of pressure pg over the surface of 
the liquid, the condition at the surface becomes 
W = [Vo + + r)] + C-^] + ~ {Vo + + f)] 
+ 
V,- 
ih 
+ const. . {44'"') 
Equating to zero the non-periodic terms, we obtain the well-known condition for 
steady motion 
[Vq + + 2/^)] + const. = 0. 
Here 
where 
Vq = const. — 1 [ A (a;2 -b i/) + Cz^] 
A = 477py ^0 (4^ + 1)[ 
dC 
C = 47rpy Co {Co^ + l)|^ ' jy ^ = dTrpy cosec^ a cot 
u ^(C) 
= 47rpy cosec® a cot a tVtA 
h (W 
S\ (Co) 1^ 
Pi (Co) J 
(46), 
(47), 
y being the constant of gravitation, and being put equal to unity if the density is 
expressed in astronomical units. 
From (45) we have, in the usual manner, 
(A - 0,2) (^2 q. ^ 2 ) q_ c ,2 = {{x^ + /)/(^,/ + 1) + zyCo^} 
whence 
c.®=A-C4//(Co^ + l) 
= 4W&{<l"(£))V(&)-Pi(£o)2l(£j)} • • ■ 
(«). 
(«). 
which can also be put in the form 
= 47rpy ^0 (1 (3^o^ + 1) cot ^ 
= 47rpy Coqz (Co). 
fCol 
(50). 
From (44'''), (45) the boundary-condition for the oscillations is 
M = M + 8^0 Af {Vo -f W (^^ -f r)] + 
Vo, 
P2 
pj 
. . (51). 
